DR. NARINDER PURI ANDDR. MICHAEL WEINLESS

ON VEDIC MATHEMATICS

N. Puri and M. Weinless, Vedic Mathematics: The Cosmic Software For The Cosmic Computer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

T. Brooks and K. Brooks, Vedic Mathematics: Taking the Anxiety Out of Math . . . . . . . . . . . . . . 8

J. Hamilton and E. Norlyk, Vedic Mathematics: “To Be Learned With Smiles” . . . . . . . . . . . . . 11

VEDIC MATHEMATICSThe Cosmic Software For

The Cosmic Computer

Dr. Narinder PuriUniversity of RoorkeeUttar Pradesh, India

Dr. Michael WeinlessMaharishi International University

Fairfield, Iowa, USA

Presented at the National Council of Teachers ofMathematics Annual Conference, Chicago, IL, 1988

Introduction

Vedic Mathematics offers a new holistic approach to mathematics and to mathe-matics education. Its range extends from the most concrete values of numerical compu-tation to the most abstract aspects of the dynamics of intelligence.

The computational methods of Vedic Mathematics have been found to be easier tolearn, faster, and more enjoyable than conventional methods. The intrinsic flexibility ofthe Vedic system of calculation develops the students’ creativity and clarity of mind.

The system of Vedic Mathematics is derived from the Vedic tradition or knowl-edge, recently revitalized and reinterpreted by Maharishi Mahesh Yogi in the form of amodern systematic science, known as Maharishi’s Vedic Science. According to Mahar-ishi, at its deepest level, Vedic Mathematics is the exploration and exposition of the self-interacting structure of pure consciousness, which is the orderly basis of all phenomenain nature. By connecting mathematics to its source in the field of pure consciousness,Vedic Mathematics has the potential to bring fulfillment to all areas of mathematics andto solve the current crisis in mathematics education.

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The Sutras of Vedic Mathematics

All the computational methods of Vedic Mathematics are based on sixteen sutras(aphorisms) that are directly derived from the Atharv Ved, one of the primary texts of theancient Vedic literature of India. This mental computational system was brought to lightin recent times by the Shankaracharya of Puri, Swami Bharati Krishna Tirtha.

On one level, each sutra aphoristically describes the actual computational methodto be used for the various types of mathematical problems. On a deeper level, however,each sutra may be seen as a refined formula for producing a high degree of coherence andorder in brain functioning, facilitating the rapid and precise solution to the mathematicalproblem.

The coherently functioning brain, with its billions of neurons and interconnectionsis, in Maharishi’s words, a “cosmic computer.” Through proper programming it can ac-complish anything. The sutras of Vedic Mathematics are the “cosmic software” that createthe ability to compute most rapidly and precisely.

Applications to Mathematics Education

Vedic Mathematics offers a new approach to resolving the current crisis in math-ematics education. It is not simply a collection of new computational techniques; ratherit provides an entirely different approach to mathematical computation, based on patternrecognition.

The conventional methods of mathematical computation consist of rigid, some-times monotonous procedures that are uniformly applied to all problems of a given type.The result is often boredom, strain, and a lack of enjoyment for both student and teacher.

By contrast, the techniques of Vedic Mathematics allow for constant expression ofthe student’s creativity. First, for any type of problem, there are always several differentapproaches: at least one general technique, applicable to all cases, and also a number ofspecial techniques, applicable to cases having a particular pattern. The general techniqueis always faster and easier than the conventional technique, while the techniques for spe-cial cases extraordinarily fast, yielding solutions to complex problems, often at the speedof just writing down the successive digits of the answer. Second, at each stage of a par-ticular technique, there are usually options with regard to the next step. This flexibilityin the method keeps the mind lively and alert and cultures the ability to quickly discoverthe path of least action on the way to the solution.

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In addition, the techniques of Vedic Mathematics have been found to be easier tolearn than conventional methods. Computational complexities are significantly reducedby using digits and their complements. And finally, unique, independent cross-checkingmethods make the system less prone to error.

In summary, then, the techniques of Vedic Mathematics not only provide a quickand accurate solution to the problem at hand, but also develop clarity of mind and intu-ition. In this way, Vedic Mathematics succeeds in fulfilling the two major goals of mathe-matics education: to teach students procedures for solving specific mathematics problemsand to develop clear and logical thinking.

In teaching Vedic Mathematics, Dr. Puri has found that students derive great en-joyment from it, even if they previously disliked mathematics. The problem of “math anx-iety,” which confronts mathematics educators around the world, simply does not arise.The Vedic mathematics research group in London has successfully applied Vedic Math-ematics at both primary and secondary school levels. The Indian Ministry of HumanResource Development recently held a workshop on Vedic Mathematics at Rajasthan Uni-versity, which was attended by mathematicians, educators, and government representa-tives. The workshop recommended that the government immediately introduce VedicMathematics into the educational curriculum of the nation.

Glimpses of Vedic Mathematics

The following elementary examples provide a glimpse of the speed and simplicityof the computations provided by Vedic Mathematics.

• 99976 × 99998 = 99974|00048The answer is directly obtained in just one line using the nikhilam sutra.

• 374 − 253 + 254 − 433 + 327 = 374 + 253 + 254 + 433 + 327 = 269The answer is directly obtained by treating the numbers to be subtracted as con-sisting of negative digits, using the vinculum method.

• 252 × 254 = 14 × 256|008 = 64008

The answer is directly obtained using the anurupyena upasutra.

• 99932 = 9986|0049The square is obtained directly by using the yavadunam sutra.

• 1952 = 19 × 20|25 = 38025The square is obtained using the ekadhiken purvena sutra.

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• 988 × 996 × 995 = 979|128|240 = 979127760The product is obtained in one line using the nikhilam sutra.

• 1/39 = 0.0̇25641The answer is obtained in just six seconds using the ekadhika sutra.

• 4 yd. 1 ft. × 8 yd. 2 ft. = 32|16|2 = 37 sq. yds. 56 sq. ft.The answer is obtained using the urdhava sutra.

• 202892 = 203112= 4|0|12|4|5|6|5|2|1 = 412456521 = 411643521

The result can be directly obtained by working from either left to right or right toleft in four alternative ways using dwandwa yoga.

These examples provide just a glimpse of the most elementary applications ofVedic Mathematics. More sophisticated application include computational aspects of lin-ear algebra, curve fitting, evaluation of transcendental functions, solution of cubic andhigher order algebraic equations, and the solution of certain types of nonlinear problemsin analysis.

The Foundations of Vedic Mathematics

Computational techniques are just one aspect of the complete approach of VedicMathematics. Maharishi’s Vedic Science locates the deepest value of Vedic Mathematicsin the mathematical structure of consciousness itself. The founders of the Vedic traditiondiscovered the capability of the human mind to settle into a state of deep silence whileremaining awake, and therein to experience a completely unified, simple, and unboundedstate of awareness, called pure consciousness. In this experience, Maharishi explains, theconscious mind becomes identified with an infinite, all-pervading unified field, an eternalcontinuum underlying all existence.

By gaining familiarity with the silence of pure consciousness, one develops theability to cognize within that silence an eternal and unmanifest “fabric” or “blueprint” ofall the laws of nature; this fine fabric of activity is the Ved. The Vedic seers gave expressionto the self-sufficient, infinitely dynamic, self-interacting qualities of this unified state ofawareness; and they articulated the dynamics by which it sequentially gives rise, throughits own self-interacting dynamics, to the field of space-time geometry, and subsequentlyto all the diverse forms and phenomena that constitute the universe. These expressionsand articulations constitute the Vedic Literature.

At its most profound level, Vedic Mathematics is the mathematics of the Ved, thestructure of pure knowledge; it reveals the precise mechanics by which the infinite, ab-solute reality of pure intelligence knows itself and thereby creates, from within itself, all

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quantified values and relationships. On this basis, Vedic Mathematics is able to portraythe quantification of all levels of intelligence (pure mathematics) and also to describe thedynamics of intelligence found in all levels of nature’s functioning (applied mathematics).

Maharishi describes the Ved, within the field of pure consciousness, as existing onthe level of undivided unity. Within this unity two values, existence and intelligence, canbe located. It is the intelligence value that knows itself and thereby conceptually createsthree values within itself—the knower, the process of knowing, and the known. Thus,one becomes two, and two becomes three. Through the further interaction of these threevalues, the infinite diversity of the universe is created from the sequential quantificationof unity. This unfoldment is portrayed with perfect precision by Vedic Mathematics.

According to Vedic Science, all values of relationship and transformation, both ab-stract and concrete, are simply expressions of the self-interacting structure and dynamicsof an underlying, totally unified field of pure intelligence. The unlimited variety of re-lationships and transformations, ranging from the most abstract areas of modern mathe-matics to the most concrete phenomena of nature, can thereby be described in terms of theinner relationships of pure consciousness. Vedic Mathematics provides this description.

This general principle is familiar to modern mathematicians. The unified founda-tion of modern mathematics is provided by set theory, the mathematics of the infinite. Atthe foundations of set theory are mathematical principles that directly describe the natureof an abstract, infinitely dynamic field of intelligence. All of mathematics is described assequentially unfolding from the internal dynamics of this field of pure intelligence. Forexample, all sets are sequentially created from the unmanifest point value of the null set.This self-referral nature of the field of intelligence at the basis of set theory is expressed ata deeper level by the reflection principle, which underlies the mathematical descriptionof the different quantified values of infinity provided by the theory of large cardinals.

In Vedic Mathematics, all the fundamental principles are derived from direct ex-perience of the unmanifest source of orderliness within the unified field or pure con-sciousness. Vedic Mathematics is, therefore, the mathematics of the most fundamentalstructures of natural law. This is its unique stature and contribution to the field of knowl-edge.

Invitation to All Mathematicians

The computational applications of Vedic Mathematics are only the first step in anongoing research program. All areas of modern mathematics, both pure and applied,fall within the scope of Vedic Mathematics. We are just beginning to tap the enormouspotential, of this more fundamental and complete approach to knowledge,

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By connecting mathematics to its source in the field of pure consciousness, VedicMathematics has the potential to bring fulfillment to all areas of mathematics. All math-ematicians and mathematics educators are invited to join this research program, to applythe sutras of Vedic Mathematics to all areas of mathematics, and to enjoy the full creativeexpression of mathematics for the benefit of mankind.

For more information write to: Institute of Vedic Mathematics, Maharishi InternationalUniversity, Fairfield, Iowa 52556.

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VEDIC MATHEMATICSTaking the Anxiety Out of Math

Tom and Kathy BrooksMaharishi International University

Fairfield, Iowa, USA

MIU News, Spring 1988

“Vedic Mathematics makes math a playful subject—a subject to be learnedwith smiles,” said Dr. Narinder Puri, a professor at the University of Roor-kee in Uttar Pradesh, India, and a leading expert in Vedic Mathematics.Dr. Puri has authored a book on the subject and along with several En-glish and American mathematicians and mathematics teachers, is reform-ing mathematics education using the Vedic method.

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“Mathematics doesn’t have to cause anxiety, but can be blissful and evolutionary,”said Dr. Narinder Puri during his April 14 presentation to over 1000 members of the MIUcommunity.

Dr. Puri, Professor of Civil Engineering at the University of Roorkee, UttarPradesh, India, and a leading exponent of Vedic Mathematics, visited MIU as part of aworld tour to introduce Vedic Mathematics to modern science based education.

“Mathematics education aims to culture clear and precise thinking,” says Dr. Puri.“But in the conventional approach to teaching basic math skills, calculations are often astrain. This results in ‘math anxiety’ among students and a shunning of the subject. Testscores have declined, and we are now talking about a ‘crisis’ in math education. VedicMathematics, by giving easy procedures in tune with natural law, makes the process ofdoing math enjoyable and enriching.”

The computational methods of Vedic Mathematics are based on sixteen sutras(aphorisms) that are directly derived from Atharva-Veda, one of the four main branchesof ancient Vedic literature.

On one level, each sutra aphoristically describes the computational method to beused for vanous types of mathematical problems. On a deeper level, however, each sutramay be seen as a refined formula for producing a high degree of coherence and order inbrain functioning, facilitating the rapid and precise solution to the mathematical problem.

MIU’s founder, Maharishi Mahesh Yogi, who recently met with Dr. Puri at theWorld Capital of the Age of Enlightenment in India, explained that Vedic Mathematics,through its sutras, helps to create a general state of awareness, which is then focused ona particular point (a specific problem). Cultivating the ability to maintain this wholenesswhile focusing on the parts of knowledge helps one to live more in accord with naturallaw.

According to Dr. Puri, this new approach is not just a collection of new teachingmethods but rather an entirely different way of doing mathematics. The conventionalmethods of mathematical computation consist of rigid, sometimes monotonous proce-dures that are uniformly applied to all problems of a given type.

In contrast, the techniques of Vedic Mathematics allow for constant expression ofa student’s creativity. At each stage, there are options with regard to the next step. Thisflexibility in the method keeps the mind lively and alert and cultures the ability to spon-taneously discover the path of least action on the way to the solution.

Vedic Mathematics not only provides a quick and accurate solution to the problemat hand, but simultaneously develops clarity of mind and intuition, emphasizes Dr. Puri.In this way, Vedic Mathematics fulfills the two major goals of mathematics education:to teach students procedures for solving specific mathematical problems and to developclear and logical thinking.

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Students are also able to learn Vedic Mathematics computational methods four tofive times faster than the conventional system. During his visit to MIU, Dr. Puri visitedMaharishi School of the Age of Enlightenment (grades K–12), where he demonstratedthe ease of mental calculation of large multiplication problems. He could write a one-line solution to the problems before the students could even enter the numbers into acalculator.

Accompanying Dr. Puri on his tour was Dr. Michael Weinless, Chairman of theDepartment of Mathematics at MIU and a distinguished mathematician and educator.Dr. Weinless explains that the basis of Vedic Mathematics, as described by Maharishi’sVedic Science, is in the mathematical structure of pure intelligence, which is the orderlybasis of all phenomena in nature.

“Vedic Mathematics, at its most fundamental level, reveals the precise mechanicsby which the infinite, absolute reality of pure intelligence knows itself and thereby creates,from within itself, all quantified values and relationships,” states Dr. Weinless.

“All areas of modern mathematics, both pure and applied, fall within the scopeof Vedic Mathematics. We are just beginning to tap the enormous potential of this morefundamental and complete approach to knowledge,” Dr. Weinless points out.

MIU is launching a whole new phase of mathematics research which examines theapplication of this knowledge to all areas of science. The introduction of Vedic Mathemat-ics not only influences the future of mathematics education, but current research indicatesthe principles of Vedic Mathematics can be applied not only to the basic areas of math-ematics, including arithmetic, algebra, geometry, trigonometry, and calculus, but also tothe most abstract areas of modern mathematics, including set theory and logic, the theoryof computation, number theory, and differential equations.

Dr. Weinless notes that Vedic Mathematics, as an aspect of Maharishi’s Vedic Sci-ence, is the latest development in Maharishi’s Master Plan to Create Heaven on Earth.“Heaven on Earth means life without mistakes, where every thought and action is spon-taneously in tune with natural law,” Dr. Weinless states. “Vedic Mathematics will helpeducation deliver this, bringing to every student the fruit of all knowledge—life in fulfill-ment and perfection.” �

Copyright c© 1988 Maharishi International University [now Maharishi University of Management http:/ /mum.edu].

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VEDIC MATHEMATICS“To Be Learned With Smiles”

Joel Hamilton and Eva Norlyk

The Fairfield Source, May 1988

Dr. Michael Weinless with Dr. Narinder Puri, who taught one-day seminaron Vedic Mathematics to nearly 700 people in April.

“Does anyone have a calculator?” asks Dr. Puri. Everyone looks up at Dr. Puri,expert in Vedic Mathematics, and here to teach the basics of Vedic sutras of computation.Is he asking for a calculator?

”Yes,” continues Dr. Puri. “Does anyone have a calculator? I want to have a race.”

On the blackboard Dr. Puri whizzes through a multiplication problem of two nine-digit numbers. He works so quickly that a competing pocket size calculator is left farbehind, spitting error messages.

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“You see?” he says with a satisfied smile. “Vedic Math is verry easy!”

Dr. Narinder Puri, professor at the University of Roorkee, Uttar Pradesh, India,recently visited MIU to teach a seminar on the computational techniques of Vedic Math-ematics. Nearly 700 people attended his one-day seminar in the Maharishi PatanjaliGolden Dome of Pure Knowledge on Saturday, April 16. The seminar was part of a con-ference tour of universities throughout North America and Europe.

Based on 16 sutras derived from the Atharv Veda, the computational methods ofVedic Mathematics reportedly will enable students to breeze through complicated com-putations with the effortless speed of a supercomputer.

“Vedic Mathematics is easier to learn, faster to use, and less prone to error thanconventional methods,” says Dr. Michael Weinless, Chairman of the Department of Math-ematics at MIU. “Furthermore, the techniques of Vedic Math not only enable the studentto solve specific mathematical problems, they also develop creativity, logical thinking,and intuition.”

According to Dr. Puri, Vedic Math is the math software for the “cosmic computer,”a way for the human nervous system to make use of the computing power of nature.

“Numbers are your friends,” he says. “Once they become your friends, you’ll loveplaying with them. Vedic Math makes mathematics a playful subject, to be learned withsmiles.”

Many course panicipants seemed to harbor hopes that the sutras would be likemagic words to make all math problems disappear, leaving only answers in their place.But the sutras’ magic was on a very practical, mechanical level. These 16 Sanskrit apho-risms, only 120 words in all (“so brief and crisp they are”), serve as mnemonic devices,each one bringing to mind a class of mathematical operations which can be applied invarious situations.

Dr. Puri explained how Vedic Math, as much as possible, breaks up complex prob-lems into simple short operations. A common device is to convert the larger digits, 6 to9, into the smaller, easier-to-handle digits, 1 through 5. Dr. Puri calls the larger numbers“mischievous,” difficult to manipulate. Their complementary numbers are called vincu-lum numbers, and they require using as base a power of 10 (10, 100, 1000, etc.).

The reason for this is simple, according to Dr. Puri. “If you want to measure howtall a man is, you can use a ruler and start from his feet. But if he is a very tall man, itmight be easier to start from the ceiling and measure down.”

The vinculum number is the base number plus or minus some quantity. For exam-ple, 9 becomes 10 − 1, written in Vedic Math as 11. This complementary number can thenbe handled in a host of ways to speed and simplify calculations.

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Dr. Puri spent a good part of the Saturday workshop explaining the second sutra,Nikhilam Navtashcaramam Dashata. A simple translation of this is “All from 9 and lastfrom 10,” although according to Dr. Puri the English translation captures none of theexpansiveness and depth of the Sanskrit meaning. In India, Maharishi told him that hecould derive all the sutras from this one.

Using the Nikhilam sutra, Dr. Puri showed simple steps to multiply large numbersand to speedily create multiplication tables. Later he demonstrated how another sutra,Urdhave-Tiryagbhyam (meaning “vertically and crosswise”), can also be used to multiplylarge numbers easily. A third sutra, translated as “By one more than the previous one,”Dr. Puri considers an “upa-sutra” (subordinate sutra, a corollary) to the Nikhilam sutra.He showed its use in writing the decimal equivalents of large awkward fractions such as1

13 , 17 , and 1

29 .

“Vedic Math doesn’t want you to get bored and exhausted.” says Dr. Puri. “Effort-less effort is the real content of Vedic Math.”

Dr. Puri first met Maharishi early this year at Maharishi Nagar. Maharishi wasso taken with Dr. Puri and the potential of Vedic Math that he immediately suggested aworld tour to promote it.

Vedic Mathematics is more than the computational techniques taught by Dr. Puri.Its range extends from the applied values of math to the most abstract aspects of the dy-namics of pure intelligence. According to Maharishi’s Vedic Science, Vedic Mathematics,at its deepest level, reveals the mechanics by which pure consciousness knows itself. Theinfinite diversity of the universe emerges from the conceptual quantification of unity intothe three values of knower, known, and process of knowing–rishi, devata, and chhandas.

“The expression of wholeness unfolding into quantified values can be analyzedin mathematical terms,” says Dr. Michael Weinless. “All relationships and transforma-tions, from the most abstract areas of modern mathematics to the most concrete levels ofnature’s functioning, can be described in terms of the inner dynamics of pure conscious-ness.

“In Vedic Mathematics all the fundamental principles of math are derived from thedirect experience of the unmanifest source of orderliness within the unified field of pureconsciousness. Vedic Mathematics is the mathematics of the most fundamental structuresof natural law.”

Textbooks on Vedic Mathematics are being written and translated into 30 lan-guages, and correspondence and video-taped courses are now available through a privateorganization in India. MIU’s departments of Mathematics and Continuing Education areplanning more seminars and conferences.

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Dr. Puri told a story of the monumental work of Swami Bharati Krishna Tirtha, therecent Shankaracharya of Puri, the easternmost of the four seats of the Shankaracharyatradition. The Shankaracharya had spent eight years distilling the essence of Vedic Math-ematics from the Atharv Veda. His exposition and commentary on Vedic Math ran to 16volumes.

When the work was done, the manuscripts were sent off to be edited and pub-lished. Somewhere along the way, the manuscripts were destroyed. The Shankaracharya’sdistraught assistants had to break the news to him.

“Swamiji! Something terrible has happened! Your precious work on Vedic Mathhas all been destroyed!”

“So it’s lost?” said the Shankaracharya. “It doesn’t matter.”

“But the whole knowledge is lost!” said the breathless assistant, unconsoled.

“Why are you upset?” asked the Shankaracharya in reply. “The knowledge isnever lost. The Vedas are always there. It just goes behind a curtain sometimes. It willcome again.”

The curtain’s pleats are parting once again on Vedic Mathematics. �

Copyright c© 1988 The Fairfield Source [now The Iowa Source http:/ /www.iowasource.com].

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